Background
The oldest surviving document demonstrating knowledge and proficiency in the false position method is the Indian mathematical text Vaishali Ganit (c. 3rd century BC). The ancient Chinese mathematical text called''The Nine Chapters on the Mathematical Art'' (九章算術) dated from 200 BC to AD 100 also mentions the algorithm. In this text, however, the example problems posed apply the false position method to linear equations only, and the solutions reached are arrived at in only one step. Leonardo of Pisa (Fibonacci) mentions false position in his book Liber Abaci published in AD 1202, following the method that he learned from Arab sources, though it was not used in the first 126 pages of Sigler's translation. The Liber Abaci's first 124 pages cite seven methods that converted rational numbers to Egyptian fraction series, four or five of which date to the time of Ahmes and the RMP 2/n table. The regula falsi method is a variant of the secant method. Given a continuous real-valued function of a real-variable, f(x), in which one wants to approximate the location of at least one root the regula falsi method proceeds after finding two values such that the function evaluated at the two values has opposite signs, i.e. if a'' and ''b are two numbers such that f(a)·f(b) < 0 then a zero of f(x) exists on the interval (min(a,b), max(a,b)). The fact that a zero of f(x) exists in the interval (a,b) is a consequence of the intermediate value theorem of elementary analysis or from topology using the lemma that the continuous image of a connected set is connected as well as the lemma that a subset of R,(with the usual topology) with more than two points is connected if and only if it is an interval. The regula falsi iterative procedure uses two estimates as in the secant method to estimate a new estimate, but the estimate which is replaced is the one for which the sign of the function is the same as the sign of the function for the new estimate. The regula falsi method is guaranteed to converge, but convergence may be extremely slow. The iterative process is continued until the halted. In the implementation given below the process is either terminated by the user, or when there is no machine representable number between the two endpoints, or when the function evaluates to zero at the new estimate. It is best to avoid using the regula falsi method for final convergence to a root. It can be used as a good starting method. The restriction that the two estimates bracket the root such that the function evaluated at the two estimates has opposite signs means that regula falsi avoids the problems associated with having local extrema since it is no longer possible to attempt to divide by zero as in the secant method. Since regula falsi requires initial estimates in which the function evaluated at the initial estimates has opposite signs, it is impossible to use regula falsi to find the zeros of strictly nonnegative functions or nonpositive functions.